Solving higher order PDEs with isogeometric analysis on implicit domains using weighted extended THB-splines

Abstract

We study the numerical solution for higher order partial differential equations defined on implicit domains. We represent isogeometric analysis on implicit domains and construct weighted extended truncated hierarchical B-splines. Specifically, the spline basis functions are classified, according to the cells occupied by their support inside the domain, using a level by level classification mechanism. A hierarchical extension is then proposed to connect basis functions with small support in the computational domain to more stable basis inside the domain. The basis functions are formulated to acquire local adaptive refinement and preserve stability to release the full approximation power. A numerical illustration is performed to solve the Biharmonic equation on a number of implicit domains with holes and reentrant corners. The error results and condition numbers of the numerical examples reflect the potential of the approach.